
The solution of the BIP (REF ) can be characterized as the unique minimizer \(\mu ^{\Delta ,y} = \operatornamewithlimits{argmin}_{\nu \in {\mathcal {P}}(X)} J^{\Delta ,y}(\nu )\)  of the following functional \(J^{\Delta ,y}: {\mathcal {P}}(X) \rightarrow \mathbb {R}\)  (cp. e.g. [1]}):
\( J^{\Delta ,y}(\nu ):={\mathrm {KL}}(\nu ||\mu ) + \int _X \Phi ^{\Delta ,y}(u) \, d\nu (u),\) 
